I'm undergraduate student in probability theory (and its applications). There are lots of different and definitely good text on standard, functional analysis-based approach, but I'm interested in alternative approaches - maybe, some more algebraic variants. Could you name some ideas/papers/texts about this? I'm especially interested in the ones those can be used in applied problems (such as financial mathematics or something). I've surely saw some approaches in Wikipedia, but I have absolutely no idea of using them in practical problems.
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1$\begingroup$ Perhaps Whittle's Probability via Expectation? It depends at what level you want to study probability theory - Williams's Probability with Martingales is an excellent book if you want to start thinking about stochastic processes in discrete time, but there is no avoiding a certain amount of work with measures and convergence theorems. $\endgroup$– Yemon ChoiCommented May 21, 2011 at 8:05
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1$\begingroup$ Perhaps you could give an example of the kinds of text you are not after - e.g. "this is a good book, but I want something which approaches this part differently" $\endgroup$– Yemon ChoiCommented May 21, 2011 at 8:07
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3$\begingroup$ If you just want to get some probabilistic intuition, then something like Grimmett and Stirzaker might be worth a look. $\endgroup$– Yemon ChoiCommented May 21, 2011 at 8:08
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1$\begingroup$ It's possible to base a theory of probability on Von Neumann algebras. This construction is known as free probability, and has applications to quantum mechanics. See Terry Tao's book on random matrices. There's a free preprint on his website. $\endgroup$– Simon LyonsCommented May 21, 2011 at 11:45
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4$\begingroup$ @Simon: as someone who's dabbled in both, I am not sure that starting with free probability is a good way to learn about classical probability. $\endgroup$– Yemon ChoiCommented May 21, 2011 at 18:45
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2 Answers
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Yuri, you may have a look at the introductory probability book by Henk Tijms, Understanding Probability. An excellent book having the nice feature that simulation is used throughout to develop probabilistic intuition.
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Peter Whittle's "Probability via expectation" is a very nice book. In taking expectation (rather than probability measure) as a starting point, the text manages to
- Avoid measure theory almost entirely
- Get to interesting applied problems very quickly
Though Whittle (probably deliberately) never says this, the approach here is essentially the `Daniell integral' approach to integration theory, cast in a probabilistic context. The text is (in my opinion) spectacularly well written, both in terms of the choice/order of material, and in the quality and warmth of the written English. I cannot think of many books of this type (i.e. one advocating an 'alternative' approach to the development an established theory) which cover so much material quite so masterfully whilst remaining so focused and true to their founding premise.
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$\begingroup$ Just spotted that Yemon has already mentioned this one in the comments :) $\endgroup$– DCMCommented Apr 7, 2019 at 16:09